LOL From the time my kids and grandkids learn to walk they are told no touching no running and no asking for anything. Then when I tell them they may choose something within reason they know what I mean and I haven't ever had a problem with mine in the store. When my kids were little my DH always went along and stayed in the car. If the kids misbehaved they had to go sit in the car with dad - not a place they wanted to be when they were in trouble! LOL
Also I was a foreign language major in college and I talked to my kids in Spanish when I didn't want to be overheard. There were few people then who understood me. I could correct them without sounding like I was. They knew not to act up in public. It just wasn't allowed. No question about it! They learned to be polite and respectful and that lesson has helped them in life.
I also don't let the grands act up either. I can find a good spot for a timeout if need be as Zach can attest to!! He has had to sit many times. One time I was shopping for fabric and I had Devon (then 5) and Rachel(6 at the time) and they got too carried away with their game and kept stepping in front of people. So I had them sit by my feet on the carpet as I looked at fabric. Worked just fine. They continued their conversation and passed the time without bothering any other shoppers.
I just hate to see people let their kids run wild in a store!!